Translator: Jim Taylor Reviewer: Denise RQ
Does anyone even know how to operate one of these things?
(Laughter)
I'm expecting buttons.
When I tell people I'm a card-carrying mathematician,
sometimes, they believe I can operate this thing like nobody's business.
I can, in fact, make it play like Jimmy Hendrick's used to play the guitar.
Check it out.
(mock rock music guitar playing)
But this is actually more my style.
(mock hard rock music playing)
I have no use for this thing. In fact, I'd like to give it away.
Does anyone need one of these things?
(Laughter)
I'm serious.
Come on up. It's right here.
It's yours. Take it, if you want it.
Other times when I tell people what I do for a living,
I get this response [Oh, I hated math.]
which is surprising to me where the typical decorum is politeness.
Anyway, it's not that offensive to me,
and I do understand that by 'hated'
they don't mean they used to hate math, and now they're OK with it.
(Laughter)
I know that they mean that they hated it, and they're done with it,
and they'll never have anything to do with it again.
That's not exactly a personal affront to me.
The offense I take in it is that it's a reflection of my profession.
After all, I went through the experience that this person I'm talking to went with.
I went and saw the movie that is the math curriculum.
Which, by the way,
it would be called "The Big Foist" if it was actually a movie.
The taglines would be unambitious and offensive.
Offensive because it's offensive
to any student who actually wants to learn something
and doesn't want to be treated like they're stupid.
It's unambitious because if you're a teacher,
you actually have nothing but a goal of having your students take a test;
whether they pass or not, I claim that's not an issue.
Think about it.
I showed up to this movie, and this is what happened.
Wax on, wax off, wax on, wax off.
Wax on! Paint the fence. Paint the house.
Sand the floor. Wax on! Wax off!
And then I'm done.
(Laughter)
(Applause)
I could've gone out into the world and got a job.
After all, I did pretty well on those exams that I took,
and speaking of exams, they're very tricky things.
Consider this prompt for example.
I was involved in a workshop to help construct the core curriculum,
and a good teacher, like a good lawyer, knows the answer to any question
that you're going to ask.
But a really good teacher knows
every possible response to a question they ask,
and the depth and breadth that that response implies
about their knowledge that they're testing for.
So, here was a prompt that we had to consider.
The response that we had to think about was this one.
I can't remember if we were told by the person running the workshop
or if we figured it out,
but what I came away with
was the student apparently used the number line
as the vinculum for the fraction
which is doing exactly what the prompt says.
(Laughter)
Looking back, I honestly wonder about how much is actually evinced
in these tests that we give.
Numbers aren't a line;
why should anyone know what a number line even is?
Is this actually testing
whether I know that 3/4 lies between 1/2 and 1?
Why don't you just ask me that
instead of telling me to do something like this.
Another prompt that everyone who I've ever asked
has been on the receiving end of or at least of a prompt like this ...
And we all know that the answer that's desired is E.
But think about it; the real answer is G.
But that's never there.
That's an answer for every possible prompt like this, every single one.
Because we have no idea what's causing that sequence of numbers.
It could be the number of goos that their baby has uttered
since they woke up.
During the first minute, well, they were sleeping.
The second minute: five goos.
The next minute, ten - because they're getting awake I guess.
I don't know how many goos are going to happen next.
It certainly has an upper bound. It's not increasing to infinity.
Goo, goo, goo, goo, goo, goo! I actually don't know.
But having done well on these tests and not ready for the real world,
certainly not really wanting to get a job, I went back to college.
Because I was told when I was in high school
that when a teacher actually gave an honest response,
"What is this crap good for?" -
We didn't usually say crap -
"What is this stuff good for?",
the only response I came to respect was the one that said,
"Well, you're probably going to use it in college,"
or the better one was, "Well, you're going to use it on the test later."
So I went back to college thinking,
"I've been prepped for 14 years to come to this.
I want to learn more math. I don't know why.
I want to do some more substantive learning."
So I went back to college,
and wax on, wax off, wax on, wax off, wax on
minus three, arbitrary constant.
I nailed the arbitrary constant, got my degree, and went on.
I got a real job and suffered.
Suffered under the stymying, terrible,
day-to-day, creativity-killing job
that was just making me money.
I got caught up in climbing the corporate ladder, I'll admit.
But I didn't know
that I didn't know anything about mathematics
until I went back to graduate school and was knee-deep in it
facing real challenges, facing problems
that forced me to be creative, to be clever,
and by the way, mathematics is a place where creativity is definitely needed.
And that creativity is put to the test
in a way that's different from any other art form.
Yeah, it's an art form.
Why it's in the college of science? I have no idea.
But I also realize that these misconceptions
that that curriculum generated are present in colleagues
that I have on my university.
For example, administrators.
For example, people that might control
what the math department does, what it needs to teach.
Let me give you an example. Here's a story.
There are these departments in the college of science:
Mathematics, Chemistry, Geology, Physics, and Biology.
These are the logos that are currently in use.
We were asked to revamp our logo,
not sure why, maybe because it wasn't e-sexy, or iHot, or whatever,
or hashtag cool, or something.
But we were asked to reevaluate our logo which is the one in the center there
which is of course, an M on its side, which stands for math
because that's what mathematicians do: is they obfuscate things.
That's all we're here to do.
In fact, here's a quote from a physics colleague,
someone in the Physics Department who was explaining a problem and said -
this is a quote -
"And then some mathematicians got a hold of the problem,
and it was completely incomprehensible."
(Laughter)
My internal response was,
"No, you just don't know enough math to understand it.
There's a likelihood
that your problem was actually solved by those mathematicians,
but I can't blame you for not knowing enough math.
There's no sin in that.
Our curriculum, our universities optimize the amount of math you don't take.
And they stop you at an arbitrary point which is typically ridiculous."
People's last experience with mathematics is something terrible,
filled with inane stuff they will never use
unless they go on to calculus, but not everyone goes on to calculus.
So, we were actually recommended ...
Someone was hired to come up with a new logo,
I guess because mathematicians certainly couldn't make them sexy enough.
So, this was the logo that was recommended to us.
(Laughter)
The damn calculator.
So, this started a flurry of emails that I still have - this was years ago -
and I love to reread these things.
The initiating one came from our Department Head,
and he was clearly irate.
He said, "I don't know exactly
what we're supposed to do with a nine-digit calculator,
but we need to do something.
"We've got to come up with something."
So, the responses from this were fantastic.
My favorite one was,
"Well, how about a five-digit calculator with the middle one extended?"
(Laughter)
I really wish that was our logo.
But the things that actually happened
were one colleague requested this one, or suggested this one:
some logical notation.
If it were translated into English, it's "If P, then Q."
What properties guarantee what other properties?
If you have property or structure P,
then you're guaranteed to have property or structure Q.
It's a set containment relationship, so we could've used this notation as well:
everything that contains property P
is contained in the things that contain property Q.
You are guaranteed to have property Q if you have property P.
P is a sufficient condition for Q, and so on, and so forth.
I said that's fantastic, we all agreed
that was a great piece of notation for our logo,
but I went up to them and said, "What about this one?
P if and only if Q. If P, then Q, if Q, then P."
That means they're precisely the same.
That's one of the most important characteristics,
that's the bread and butter of doing mathematics - invariance.
If you ask a human a question - as opposed to an alien or a dog, I guess -
if you ask a person a question, and they don't know the answer to it,
perhaps it's a complicated one.
For example, maybe you're a senator on a jury, and you're asked,
"How much should a financial predator who preys on the elderly
pay for some of the crimes they commit?"
That's a complicated question
that you probably cannot just blurt out an answer
for which I'm not an English major.
(Laughter)
You'd probably replace that question with something
like, "How would I feel if that person preyed on my grandmother?"
The point is we often do this.
But if we train ourselves, we train our mathematical abilities
and enhance them through taking substantive, real mathematics courses,
you begin to notice
when things are actually invariant, when you can change that perspective.
If I don't like perspective P, if that's not solving my problem,
I can use perspective Q.
And sometimes, solving a problem is all about changing your perspective.
But when can you do that?
Another person said, "Why don't we use this symbol?" -
the infinity cartouche.
The reasoning was it reminds our colleagues in the college of science
that they're relegated to deal with finite stuff.
They go out into their finite fields, and weigh finite fish,
and pet finite numbers of ducks, and measure finite numbers of rocks,
in finite terms of years.
We deal with infinite quantities,
we can solve an infinite number of problems
by our powers of abstraction,
which we practice all the time in mathematics.
And this reminds me of how I think
we actually should think about mathematics.
If I were to ask you to define what a sixth sense is,
I don't know what you would say,
but I would say it is some way, some mechanism
by which we can experience things
we cannot possibly experience with the other five.
Some way to predict the future, perhaps.
But this is exactly what mathematics does:
we create models that uncannily, eerily are able to predict the future.
We can count quantities
that we could never physically - one, two, three - enumerate.
We can deduce what's going to happen in 4, 5, 50 dimensions, with mathematics.
We'd never experience those dimensions.
But then, if you'd wanted to put a little biological constraint on it
and say, "Our five senses have real estate in our brains,"
math does, too.
There's some fantastic research showing we have real estate in our brain
that's devoted to some of the substantive stuff
that we can grow into mathematics.
The mathematics understands relations, even calculus,
all kinds of things; anything in mathematics.
I think we should think of it as our sixth sense,
because if we did,
then there wouldn't be much of an excuse to not know it.
It wouldn't be OK, socially OK, to not know any mathematics,
because then, it would be clear
that you're walking around basically blinded in such a way
that you can't see things that you would see
with an enhanced mathematical skill,
you're not hearing things you could potentially be hearing.
Back to the logo.
The logo that we have actually reflects exactly the state of affairs,
it's quite poetic,
because when I took calculus 20 years ago,
it's the same calculus that I teach now.
When I was in the math curriculum 20 years ago,
it's the same curriculum that I'm in right now.
Twenty years ago,
the only way I could talk to a person at a great distance
was by using this thing that was in the shape of a banana.
It had a twirly cord that was attached to the wall and always getting tangled.
The world has changed a great deal in 20 years,
but the mathematics curriculum hasn't changed at all,
so our symbol is still this one.
And we keep feeding this monster that is devouring any chance
of students having a real experience with mathematics
and actually coming away from it with the correct sensation
that it is a creative art, a creative endeavor
that is challenging, and it develops yourself.
Every time I solved something significant as a graduate student,
I felt like I became a better person, I felt like I became smarter,
I felt like I made new connections in my brain.
I don't feel that when I do calculus or do arithmetic.
It's clear that we don't. It's largely a function of memory.
The solution to this, if it were up to me,
would be to detonate everything and start all over again,
but we could change things in a way that is not exactly easy to implement
because these types of things are not scantronable.
Take that ridiculous prompt about
telling me what the next term in the sequence is
and replace it with this prompt.
You actually have to write something,
and as a teacher, you have to read something
to assess whether the person knows what they're talking about,
how much depth of understanding of mathematics do they have,
how much flexibility do they have with the mathematics.
Our symbol should be this one
because this is what we keep doing: we keep throwing money into mathematics
to try and get people through these classes
that shouldn't even be taught.
A modest proposal from me is: don't teach those classes anymore.
And this is the point where I really have to hijack my talk
because I really didn't want to give you an idea,
I don't really have one to share
other than to blow up the mathematics curriculum and start over.
But what was that I wanted to offer you is an apology,
a deep, heartfelt apology.
I am truly sorry
that the experience that you've had with mathematics
is what it is.
Because it's a shame.
It's a negative reflection on my profession.
I try and do something about it every time I teach class,
but I know that many of you will never go back to mathematics,
and for that, I am truly sorry.
(Applause)